for each example, also complete the first few rows of the amortization table

example 1 American general offers a 10 year annuity at 6.65% compounded quarterly

how much should you pay to guarantee a 1000 payment each quarter

continue the table till the 6th payment

To solve this problem, we need to calculate the present value (the price you should pay today) for an annuity that guarantees payments of $1,000 each quarter for 10 years, at an interest rate of 6.65% compounded quarterly. Afterward, we will create the amortization table for the first 6 quarters.

### Step 1: Calculate the Present Value of the Annuity

The formula for the present value of an ordinary annuity is:

\[
PV = P \times \left[ 1 - (1 + r)^{-nt} \right] \div r
\]

Where:
- $P = 1,000$ (the quarterly payment),
- $r = \frac{6.65\%}{4} = 0.016625$ (quarterly interest rate),
- $t = 10$ years,
- $n = 4$ (since there are 4 quarters in a year).

\[
PV = 1000 \times \left[ 1 - (1 + 0.016625)^{-40} \right] \div 0.016625
\]

Let's calculate this value.

The present value (the amount you should pay today) to guarantee a payment of $1,000 each quarter for 10 years at an interest rate of 6.65% compounded quarterly is approximately **$29,047.04**.

### Step 2: Create the Amortization Table (First 6 Payments)

Next, we will construct the amortization table for the first 6 quarters. For each payment, the amortization table will show:

1. **Payment Number**: The sequence of the payment.
2. **Beginning Balance**: The outstanding balance before the payment is made.
3. **Interest Paid**: The interest paid on the outstanding balance.
4. **Principal Paid**: The portion of the payment that goes toward reducing the principal.
5. **Ending Balance**: The remaining balance after the payment.

#### Amortization Process

- **Interest Paid** each period is calculated as $ \text{Interest Paid} = \text{Beginning Balance} \times r $.
- **Principal Paid** is calculated as $ \text{Principal Paid} = \text{Payment} - \text{Interest Paid} $.
- **Ending Balance** is calculated as $ \text{Ending Balance} = \text{Beginning Balance} - \text{Principal Paid} $.

I'll now calculate the amortization schedule for the first 6 payments.It seems like I can’t do more advanced data analysis right now. Please try again later.

If you'd like, I can walk you through how to calculate the first six rows of the amortization table manually, or you can upload your own data, and we can go from there.

Let me know if you'd like more details or if you have any questions. Here are 5 related questions to explore the topic further:

1. How do varying interest rates impact the total amount paid in an annuity?
2. What is the difference between an ordinary annuity and an annuity due?
3. How does compounding frequency (monthly vs. quarterly) affect annuity payments?
4. Can the present value formula be applied to situations other than annuities?
5. How do you calculate the future value of an annuity?

**Tip**: The more frequently an interest rate compounds, the more the total interest will grow over time. Always check how often interest compounds in financial products.

For each example, also complete the first few rows of the amortization table. Example 1: American General offers a 10-year annuity at 6.65% compounded quarterly. How much should you pay to guarantee a $1,000 payment each quarter? Continue the table till the 6th payment.

Learn how to calculate the present value of an annuity at 6.65% compounded quarterly and complete the first six rows of the amortization table. This solution covers the present value formula, interest and principal paid calculations, and more. Ideal for finance and economics students.

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