Buying a "vacation timeshare" means buying the right to use a vacation property for a fixed period each year. Suppose that you pay $600 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 $1200 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 $1200 in 45 years (assume a 4% interest rate compounded annually). (Round your answer to the nearest cent.)

To find the present value (PV) of the $1200 bond that is redeemable in 45 years at a 4% interest rate compounded annually, we can use the formula for present value:

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

where:
- $FV$ is the future value ($1200),
- $r$ is the annual interest rate (4% or 0.04),
- $n$ is the number of years until redemption (45 years).

Now, we can substitute the values into the formula:

\[
PV = \frac{1200}{(1 + 0.04)^{45}}
\]

Calculating $ (1 + 0.04)^{45} $:

\[
(1.04)^{45} \approx 7.108
\]

Now, substitute this value back into the present value formula:

\[
PV = \frac{1200}{7.108} \approx 169.05
\]

Thus, the present value of the $1200 bond in 45 years at a 4% interest rate is approximately **$169.05**.

### Summary:
The real value of the "guarantee" is **$169.05**.

Do you want further details or have any questions? Here are some related questions you might consider:

1. How does changing the interest rate affect the present value?
2. What would be the present value if the bond were redeemable in 30 years instead of 45?
3. How does inflation impact the value of a future payment?
4. What are the advantages and disadvantages of investing in a timeshare?
5. Can you explain the difference between present value and future value? 

**Tip:** Understanding the impact of interest rates and time on investments is crucial in making informed financial decisions.

Buying a 'vacation timeshare' means buying the right to use a vacation property for a fixed period each year. Suppose that you pay $600 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 $1200 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 $1200 in 45 years (assume a 4% interest rate compounded annually). (Round your answer to the nearest cent.)

Discover how to calculate the present value of a $1200 bond redeemable in 45 years at a 4% interest rate. This detailed solution reveals the real value of a vacation timeshare guarantee and explores key financial concepts.

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