I am considering investing in a security that will pay you ​$4,000 in 26 years.

a.  If the appropriate discount rate is 9 percent​, what is the present value of this​ investment?

b.  Assume these investments sell for ​$1,325 in return for which I receive ​$4,000 in 26 years.  What is the rate of return investors earn on this investment if they buy it for ​$1,325​?

### a. Present Value of the Investment
To find the present value (PV) of an investment, we use the formula:

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

Where:
- $FV$ is the future value ($4,000),
- $r$ is the discount rate (9% or 0.09),
- $n$ is the number of years (26 years).

Substitute the given values into the formula:

\[
PV = \frac{4000}{(1 + 0.09)^{26}}
\]

Let me calculate the present value.

### b. Rate of Return
To find the rate of return $r$ for an investment where you invest $1,325 and receive $4,000 after 26 years, we use the future value formula:

\[
FV = PV \times (1 + r)^n
\]

Where:
- $FV = 4000$,
- $PV = 1325$,
- $n = 26$.

Rearranging the formula to solve for $r$:

\[
(1 + r) = \left(\frac{FV}{PV}\right)^{\frac{1}{n}} = \left(\frac{4000}{1325}\right)^{\frac{1}{26}}
\]

Then,

\[
r = \left(\left(\frac{4000}{1325}\right)^{\frac{1}{26}}\right) - 1
\]

Let me calculate the rate of return.

I will calculate both now.### a. Present Value of the Investment:
The present value of the $4,000 to be received in 26 years at a 9% discount rate is approximately **$425.57**.

### b. Rate of Return:
The rate of return investors earn if they buy the investment for $1,325 and receive $4,000 in 26 years is approximately **4.34%**.

Would you like more details or have any questions? 

Here are some related questions for further exploration:
1. How does changing the discount rate affect the present value?
2. What would be the future value if you invested the present value at the same discount rate for 26 years?
3. How does the length of time (n) impact the present value of long-term investments?
4. What happens to the rate of return if the future value increases?
5. How would inflation affect the real value of the future cash flows?

**Tip:** The longer you wait to receive a cash flow, the more significant the impact of the discount rate on its present value.

I am considering investing in a security that will pay you ​$4,000 in 26 years. a. If the appropriate discount rate is 9 percent​, what is the present value of this​ investment? b. Assume these investments sell for ​$1,325 in return for which I receive ​$4,000 in 26 years. What is the rate of return investors earn on this investment if they buy it for ​$1,325​?

Learn how to calculate the present value of an investment of $4,000 over 26 years at a 9% discount rate. Also, find out the rate of return on an investment bought for $1,325 with a payout of $4,000 in 26 years. This example covers key finance concepts such as discounting future cash flows and exponential growth, suitable for college-level students.

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