A certain perpetuity pays the holder $900 per month. If the money is invested at a rate of 6% annually (compounded continuously), what is the present value of the perpetuity?

To determine the present value of a perpetuity, where the payment is continuous, we can use the formula for the present value of a perpetuity that pays continuously. The formula is:

\[ PV = \frac{P}{r} \]

Where:
- $ PV $ is the present value of the perpetuity.
- $ P $ is the payment per period.
- $ r $ is the interest rate per period.

Given:
- The payment $ P $ is $900 per month.
- The annual interest rate $ r $ is 6%, which is 0.06 in decimal form.

Since the payments are made monthly and we need the interest rate in terms of months for continuous compounding, we adjust the annual interest rate to a monthly interest rate. The relationship between the annual interest rate compounded continuously and the monthly rate is:

\[ r_{monthly} = \frac{r_{annual}}{12} = \frac{0.06}{12} = 0.005 \]

So, we have:
- $ P = 900 $ dollars
- $ r = 0.005 $ per month

Now, using the formula:

\[ PV = \frac{P}{r} = \frac{900}{0.005} = 180,000 \]

Therefore, the present value of the perpetuity is $180,000.

Would you like more details or have any questions?

Here are some related questions that might interest you:

1. How does the formula for the present value of a perpetuity change if the payments are not continuous?
2. What is the difference between continuous compounding and discrete compounding?
3. How would the present value change if the interest rate were compounded monthly instead of continuously?
4. What is the formula for the present value of a perpetuity if the payments are made annually?
5. How do you calculate the present value of an annuity?
6. What are some real-world examples of perpetuities?
7. How does inflation affect the present value of a perpetuity?
8. Can the present value of a perpetuity be calculated if the interest rate is variable?

**Tip:** Always ensure to adjust the interest rate to match the frequency of the payments when calculating present values.

Learn how to calculate the present value of a perpetuity paying $900 per month, compounded continuously at an annual rate of 6%. Understand the formula PV = P / r and its application in finance. Suitable for advanced learners and financial professionals.

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