American General offers a 13​-year annuity with a guaranteed rate of 6.85​% compounded annually. How much should you pay for one of these annuities if you want to receive payments of ​$1000 annually over the 13 year​ period?
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Part 1
How much should a customer pay for this​ annuity?
​$
  
enter your response here ​(Round to the nearest​ cent.

To determine how much a customer should pay for the annuity, we need to find the **present value of an annuity**. The formula for the present value $ PV $ of an ordinary annuity is given by:

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

Where:
- $ P $ is the annual payment (in this case, $1000),
- $ r $ is the interest rate per period (6.85% compounded annually, so $ r = 0.0685 $),
- $ n $ is the number of periods (13 years).

### Step-by-Step Calculation:

1. **Identify values:**
   - $ P = 1000 $
   - $ r = 0.0685 $
   - $ n = 13 $

2. **Substitute the values into the formula:**
   \[
   PV = 1000 \times \frac{1 - (1 + 0.0685)^{-13}}{0.0685}
   \]

3. **Calculate $ (1 + r)^{-n} $:**
   \[
   (1 + 0.0685)^{-13} = (1.0685)^{-13} \approx 0.4670
   \]

4. **Plug into the formula:**
   \[
   PV = 1000 \times \frac{1 - 0.4670}{0.0685}
   \]
   \[
   PV = 1000 \times \frac{0.5330}{0.0685}
   \]
   \[
   PV \approx 1000 \times 7.7803
   \]
   \[
   PV \approx 7780.30
   \]

### Conclusion:
The customer should pay approximately **$7,780.30** for the annuity.

Would you like more details on the calculation or have any other questions?

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Here are 5 related questions to consider:
1. How does the interest rate affect the present value of an annuity?
2. What happens to the annuity value if payments are made monthly instead of annually?
3. How would the present value change if the interest rate were compounded semi-annually?
4. What is the future value of this annuity after 13 years?
5. How would inflation affect the real value of the annuity payments over time?

**Tip:** The higher the interest rate, the lower the present value of future payments, since the money's future value grows faster.

American General offers a 13-year annuity with a guaranteed rate of 6.85% compounded annually. How much should you pay for one of these annuities if you want to receive payments of $1000 annually over the 13-year period?

Learn how to calculate the present value of a 13-year annuity with a guaranteed 6.85% interest rate, receiving $1000 annually. This problem involves the formula for the present value of an annuity and the concept of compound interest. Suitable for students in Grades 11-12 and college-level courses.

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