American General offers a

1212​-year

annuity with a guaranteed rate of

5.155.15​%

compounded annually. How much should you pay for one of these annuities if you want to receive payments of

​$700700

annually over the

1212

year​ period?

Question content area bottom

Part 1

How much should a customer pay for this​ annuity?

​$enter your response here

​(Round to the nearest​ cent.)

To determine how much you should pay for an annuity that provides $700 annually over 12 years with a guaranteed annual interest rate of 5.15%, we need to calculate the present value of the annuity. The formula for the present value of an annuity is:

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

where:
- $ PV $ is the present value of the annuity.
- $ P $ is the annual payment ($700 in this case).
- $ r $ is the annual interest rate (5.15%, or 0.0515 in decimal form).
- $ n $ is the number of years (12 years).

Let's calculate it step by step:

1. Convert the interest rate to decimal form:
\[ r = 5.15\% = 0.0515 \]

2. Use the formula to find the present value:
\[ PV = 700 \times \left(1 - (1 + 0.0515)^{-12}\right) / 0.0515 \]

First, calculate $(1 + r)^{-n}$:
\[ (1 + 0.0515)^{-12} = 1.0515^{-12} \approx 0.5374 \]

Next, calculate $1 - (1 + r)^{-n}$:
\[ 1 - 0.5374 = 0.4626 \]

Finally, calculate the present value:
\[ PV = 700 \times \frac{0.4626}{0.0515} \approx 700 \times 8.9845 \approx 6289.15 \]

Therefore, the amount you should pay for this annuity is approximately:

\[ \boxed{6289.15} \]

Would you like more details or have any questions?

### Related Questions

1. What is the future value of an annuity and how is it different from the present value?
2. How does compounding frequency affect the value of an annuity?
3. What is the difference between an ordinary annuity and an annuity due?
4. How do you calculate the present value of an annuity due?
5. What are some common uses of annuities in financial planning?
6. How does the interest rate impact the cost of an annuity?
7. What are the tax implications of receiving annuity payments?
8. How can inflation affect the value of annuity payments over time?

### Tip

When evaluating annuity options, always consider the impact of inflation on the purchasing power of your annuity payments over the term of the annuity.

Learn how to calculate the present value of an annuity that provides $700 annually over 12 years with a guaranteed annual interest rate of 5.15%. Understand the formula and steps involved in determining the present value of annuity payments. Suitable for advanced high school students and learners interested in financial mathematics.

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