Chad just celebrated his 30th birthday. He plans to retire when he turns 61. He currently has $40,000 in his retirement plan. He will continue to add $1,110 each month to the retirement plan for exactly 31 years, starting today. On his 61st birthday, he plans to buy an annuity that will pay monthly for 16 years, with the first withdrawal starting one month after his 61st birthday (two months after their last deposit). Assume both the retirement plan and the annuity earn 4.3% compounded monthly.
a. Identify the type of the first annuity.
Ordinary Annuity / Annuity Due	Simple Annuity / General Annuity
Ordinary (END)
Due (BGN)
 Simple PY=CY
General PY ≠ CY
b. How much money will be in Chad's account the day of his 61st birthday?

Round to the nearest cent
c. Identify the type of the second annuity.
Ordinary Annuity / Annuity Due	Simple Annuity / General Annuity
Ordinary (END)
Due (BGN)
 Simple PY=CY
General PY ≠ CY
d. How large will each of the withdrawals be?

Round to the nearest cent

Let's break this down step by step, starting with each part of the problem:

### a. Identify the type of the first annuity.
- **First annuity**: Chad is adding $1,110 monthly for 31 years, starting **today**. Since the deposits start immediately, this is an **Annuity Due** (payments are made at the beginning of each period). 

Next, since the compounding is monthly (PY = CY, where PY is the payment period and CY is the compounding period), this is a **Simple Annuity**. 

Thus, the first annuity is an **Annuity Due (BGN)** and a **Simple Annuity**.

### b. How much money will be in Chad's account on his 61st birthday?

This is the value we need to calculate based on the Annuity Due formula:

\[
A = P \cdot \left( \frac{(1 + i)^n - 1}{i} \right) \cdot (1 + i) + P_0 \cdot (1 + i)^n
\]

Where:
- $ P = 1,110 $ (monthly contributions)
- $ i = \frac{4.3\%}{12} = 0.0035833 $ (monthly interest rate)
- $ n = 31 \times 12 = 372 $ (total number of payments)
- $ P_0 = 40,000 $ (initial balance)

We'll calculate this now.The total amount in Chad's retirement account on his 61st birthday will be approximately **$1,016,624.73**.

### c. Identify the type of the second annuity.
- **Second annuity**: After retirement, Chad will make monthly withdrawals, starting **one month after his 61st birthday**, which indicates that payments are made at the end of each period. This is an **Ordinary Annuity (END)**.

Since the compounding is also monthly, this is a **Simple Annuity**.

Thus, the second annuity is an **Ordinary Annuity (END)** and a **Simple Annuity**.

### d. How large will each of the withdrawals be?

We need to calculate the size of Chad's withdrawals over 16 years, with monthly payments. The formula for the payment amount $ R $ in an ordinary annuity is:

\[
R = \frac{A \cdot i}{1 - (1 + i)^{-n}}
\]

Where:
- $ A = 1,016,624.73 $ (the amount at the start of retirement)
- $ i = \frac{4.3\%}{12} = 0.0035833 $ (monthly interest rate)
- $ n = 16 \times 12 = 192 $ (total number of withdrawals)

Let's calculate the monthly withdrawal amount now.Each of Chad's monthly withdrawals from his retirement account will be approximately **$7,332.72**.

### Summary:
- **a.** The first annuity is an **Annuity Due (BGN)** and a **Simple Annuity**.
- **b.** The total amount in Chad's retirement account on his 61st birthday will be **$1,016,624.73**.
- **c.** The second annuity is an **Ordinary Annuity (END)** and a **Simple Annuity**.
- **d.** Each monthly withdrawal will be **$7,332.72**.

Would you like more details or have any questions about these calculations?

Here are 5 questions to consider:
1. What happens if the interest rate changes during the retirement phase?
2. How would increasing monthly contributions affect the final balance?
3. How would delaying retirement by a few years change the withdrawal amount?
4. What impact would a different annuity period (e.g., 20 years) have on the monthly withdrawal?
5. How does the annuity type (Due vs. Ordinary) affect the future value or withdrawals?

**Tip:** Small changes in interest rates over long periods can significantly affect the total retirement savings, so it’s essential to monitor investment performance regularly.

Chad just celebrated his 30th birthday. He plans to retire when he turns 61. He currently has $40,000 in his retirement plan. He will continue to add $1,110 each month to the retirement plan for exactly 31 years, starting today. On his 61st birthday, he plans to buy an annuity that will pay monthly for 16 years, with the first withdrawal starting one month after his 61st birthday. Assume both the retirement plan and the annuity earn 4.3% compounded monthly. a. Identify the type of the first annuity. b. How much money will be in Chad's account the day of his 61st birthday? c. Identify the type of the second annuity. d. How large will each of the withdrawals be?

Learn how to calculate the future value of Chad's retirement savings plan using annuity due and ordinary annuity formulas. We explore monthly contributions, 4.3% interest compounded monthly, and the total balance on Chad's 61st birthday. Additionally, discover how much Chad can withdraw monthly over 16 years.

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