A man aged 55 deposits $1,000 into a fund at the beginning of each year for the next 10 years.
The fund is credited with a nominal interest rate of 8% compounded quarterly.
Starting at age 65, he will receive a monthly annuity at the end of each month for the next 10 years.
During this time, the fund will earn 5% effective per annum.
Assuming that the annuity payments exhaust the fund, what is the monthly amount of his annuity?

This is a problem involving a two-phase financial situation:

1. **Accumulation Phase** (from age 55 to 65): The man deposits $1,000 at the beginning of each year into a fund that earns a nominal interest rate of 8% compounded quarterly.
2. **Annuity Payment Phase** (from age 65 to 75): After the accumulation phase, the man will receive a monthly annuity payment for 10 years, and during this time, the fund earns 5% effective per annum.

### Step 1: Accumulation Phase (from age 55 to 65)

The interest rate is **8% nominal compounded quarterly**, which can be converted to an **effective annual rate** as follows:

\[
i_{\text{effective}} = \left( 1 + \frac{0.08}{4} \right)^4 - 1 = 1.02^4 - 1 = 0.08243216 \text{ or } 8.243216\%
\]

Next, the man deposits $1,000 **at the beginning** of each year for 10 years. The future value of an annuity due (because deposits are made at the beginning of each year) can be calculated using the future value formula for an annuity due:

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

where:
- $P = 1,000$ (annual deposit),
- $i = 0.08243216$ (effective annual interest rate),
- $n = 10$ (number of deposits).

Let’s calculate this future value.

\[
FV = 1,000 \cdot \left( \frac{(1 + 0.08243216)^{10} - 1}{0.08243216} \right) \cdot (1 + 0.08243216)
\]

### Step 2: Transition to the Annuity Phase (from age 65 to 75)

After 10 years, at age 65, the man starts receiving monthly annuity payments. During this period, the fund earns a **5% effective annual interest rate**. The amount accumulated by age 65 will now be used to provide monthly payments for 10 years (120 months).

To find the monthly annuity payment that exhausts the fund, we need to use the present value of an annuity formula for monthly payments:

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

where:
- $PV$ is the amount accumulated at age 65 (from Step 1),
- $PMT$ is the monthly annuity payment,
- $i = \frac{0.05}{12} = 0.00416667$ (monthly interest rate, since 5% is the effective annual rate),
- $n = 120$ (number of monthly payments, 10 years × 12 months).

We can rearrange this equation to solve for $PMT$:

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

Substitute the value of $PV$ from Step 1 and calculate $PMT$.

Let me now compute the results for both steps.The future value of the fund after 10 years of deposits is approximately \$15,862.995.

Now, we will use this amount as the present value for the annuity payments starting from age 65, with a 5% effective annual interest rate. Let’s calculate the monthly annuity payment using the formula from Step 2.The monthly annuity payment the man will receive, starting at age 65 for 10 years, is approximately **\$168.25**.

Would you like more details, or have any questions? 

Here are 5 related questions to expand on the topic:
1. How would the results change if the interest rates were different during the accumulation or annuity phases?
2. What would happen if the man made deposits at the end of each year instead of the beginning?
3. How is the nominal interest rate related to the effective interest rate, and why is it important?
4. Can you explain how the present value of an annuity is used to determine annuity payments?
5. How does compounding frequency affect the future value of an investment?

**Tip**: When interest is compounded more frequently (e.g., quarterly vs. annually), the effective interest rate is higher, resulting in more interest accumulation over time.

A man aged 55 deposits $1,000 into a fund at the beginning of each year for the next 10 years. The fund is credited with a nominal interest rate of 8% compounded quarterly. Starting at age 65, he will receive a monthly annuity at the end of each month for the next 10 years. During this time, the fund will earn 5% effective per annum. Assuming that the annuity payments exhaust the fund, what is the monthly amount of his annuity?

Solve a real-world financial problem where a man deposits $1,000 annually into a fund earning 8% compounded quarterly, then receives monthly annuity payments over 10 years. Learn how to calculate the future value of an annuity due and the monthly payment using financial formulas. Perfect for students in financial mathematics or actuarial science.

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