A government bond is offered that pays 3% annually to the holder indefinitely. If the income received is invested at 4.3% interest (compounded continuously), what is the present value of a $10,000 bond? (Round your answer to two decimal places.)

To determine the present value of a perpetuity with a different interest rate for reinvestment, we can use the formula for the present value of a perpetuity, adjusted for continuous compounding. The key here is to find the effective rate for the continuous compounding of the reinvestment rate.

Given:
- The annual payment rate of the bond: $ P = 3\% $ of $ \$10,000 $
- The reinvestment interest rate: $ r = 4.3\% $ annually (compounded continuously)

First, we calculate the annual payment received from the bond:
\[ \text{Annual payment} = 0.03 \times 10,000 = 300 \text{ dollars} \]

The continuous compounding rate $ r $ is given as 4.3%, or 0.043 in decimal form.

To find the present value of the perpetuity using continuous compounding, we use the formula:
\[ PV = \frac{P}{r} \]

Substituting the values:
\[ P = 300 \text{ dollars} \]
\[ r = 0.043 \]

Now, calculate the present value:
\[ PV = \frac{300}{0.043} \approx 6976.74 \]

Therefore, the present value of the $10,000 bond, given the conditions, is approximately \$6,976.74.

Would you like more details or have any questions?

Here are some related questions that might interest you:

1. How does the reinvestment rate affect the present value of a bond?
2. What would be the present value if the interest rate was compounded annually instead of continuously?
3. How do you calculate the present value of a bond with a finite maturity?
4. What is the difference between a perpetuity and an annuity?
5. How does changing the bond's annual payment rate impact its present value?
6. What are the implications of a higher reinvestment rate on the present value of a perpetuity?
7. How do government bonds compare to corporate bonds in terms of interest rates and present value calculations?
8. How does the concept of present value help investors make decisions?

**Tip:** When dealing with different compounding frequencies, always ensure that the interest rate and the payment frequency are aligned for accurate present value calculations.

Explore how to calculate the present value of a perpetuity with continuous compounding. Given a government bond paying 3% annually and reinvestment at 4.3% (compounded continuously), find the present value of a $10,000 bond. Learn the formula and step-by-step calculation process.

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