Buying a "vacation timeshare" means buying the right to use a vacation property for a fixed period each year. Suppose that you pay $500 for a vacation timeshare and receive a "money-back guarantee" that at any time the company will buy back your timeshare, or if not, give you a $1000 bond. The deception, however, is that the bond is not redeemable for 45 years. Find the real value of the "guarantee"—that is, find the present value of $1000 in 45 years (assume a 5% interest rate compounded annually). (Round your answer to the nearest cent.)

To find the present value (PV) of the $1000 bond that matures in 45 years, compounded annually at a 5% interest rate, we can use the present value formula:

\[
PV = \frac{FV}{(1 + r)^n}
\]

where:
- $ FV $ is the future value of the bond, which is $1000,
- $ r $ is the annual interest rate, which is 5% or 0.05,
- $ n $ is the number of years, which is 45.

Let's substitute these values:

\[
PV = \frac{1000}{(1 + 0.05)^{45}}
\]

Now calculate:

1. $ 1 + 0.05 = 1.05 $
2. $ (1.05)^{45} \approx 9.2442 $
3. $ PV = \frac{1000}{9.2442} \approx 108.35 $

So, the present value of the "guarantee" is approximately **$108.35**.

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Would you like more details on the steps, or do you have further questions? Here are some related questions to deepen understanding:

1. What would be the present value if the interest rate were 3% instead of 5%?
2. How does changing the compounding frequency (e.g., monthly vs. annually) affect the present value?
3. What is the future value of a $108.35 investment after 45 years at a 5% annual interest rate?
4. How would the present value change if the bond were redeemable in 30 years instead of 45 years?
5. What would be the impact if there was inflation to consider alongside the interest rate?

**Tip:** When calculating the present value, the further in the future the amount is received, the less it's worth today due to compounding effects.

Buying a 'vacation timeshare' means buying the right to use a vacation property for a fixed period each year. Suppose that you pay $500 for a vacation timeshare and receive a 'money-back guarantee' that at any time the company will buy back your timeshare, or if not, give you a $1000 bond. The deception, however, is that the bond is not redeemable for 45 years. Find the real value of the 'guarantee'—that is, find the present value of $1000 in 45 years (assume a 5% interest rate compounded annually). (Round your answer to the nearest cent.)

Learn how to determine the real value of a $1000 bond redeemable in 45 years with an annual interest rate of 5%. This problem covers essential financial math concepts like present value and compound interest, providing a clear solution step-by-step. Ideal for students in Grades 11-12.

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