Final answer:
The duration of an investment is the weighted average time of each cash flow, taking into account the time value of money. We cannot provide an exact calculation without the total value or the specific dollars involved. However, the concept relies on present and future value evaluations similar to how an initial investment can grow significantly over time with compound interest.
Step-by-step explanation:
To calculate the duration of an investment, we need to understand the concept of present value and how it factors into the timing of cash flows. If an investment returns 15% in the first year, 12% in the second year, 9% in the third year, and the remainder in the fourth year, we'd need to use the present value formula to properly account for these returns over time. Without the actual dollar amounts or the total investment value to be returned in the fourth year, we can't complete a numerical calculation. However, the concept here is that the duration of the investment is the weighted average time of each cash flow, factoring in the time value of money.
Considering the provided examples, if we want to find out how much we should pay for a future cash flow where we consider the alternative investment's rate of return, we'd use the present value formula. For example, if we're looking at a future value of $1,080 a year from now, and the current rate of return is 12%, we would not pay more than $964 for that future $1,080 because $964(1 + 0.12) equals $1,080.
Similarly, the concept of compound interest demonstrates how an initial investment grows over time. For example, starting with $3,000 and assuming a 7% annual rate of return, after 40 years, the investment could grow to be 15 times larger due to the power of compounding, as shown in the calculation, $3,000(1 + .07)^40.