3. What is the value in year 20 of a $1,000 cash flow made in year 8 if interest rates are 15 percent in years 6 through 13 and increase to 18 percent in the remaining years?
To find the value of a cash flow in year 20, given the interest rates in different years, we can use the concept of present value and future value.
First, we need to calculate the present value of the $1,000 cash flow made in year 8. Present value is the current value of a future amount, taking into account the time value of money.
Using the present value formula:
Present Value = Future Value / (1 + Interest Rate) ^ Number of years
For the cash flow made in year 8, the number of years from year 8 to year 20 is 12 years.
So, the present value of the $1,000 cash flow made in year 8 is calculated as follows:
Present Value = $1,000 / (1 + 15%)^12
Next, we need to find the future value of the present value calculated above. Future value is the value of an investment at a specific point in the future, taking into account the compounded interest.
Since the interest rate changes from 15% in years 6 through 13 to 18% for the remaining years (years 14 to 20), we need to calculate the future value separately for those two periods.
For the years 6 through 13, we need to calculate the future value of the present value using 15% interest rate:
Future Value of Present Value (Years 6-13) = Present Value * (1 + 15%)^(Number of years from year 8 to 13)
For the remaining years 14 to 20, we need to calculate the future value of the present value using 18% interest rate:
Future Value of Present Value (Years 14-20) = Present Value * (1 + 18%)^(Number of years from year 8 to 20 - Number of years from year 8 to 13)
Finally, to find the total value of the cash flow in year 20, we need to sum up the future values from the two periods:
Total Value in Year 20 = Future Value of Present Value (Years 6-13) + Future Value of Present Value (Years 14-20)
And that should give you the value of the $1,000 cash flow made in year 8 in year 20, considering the given interest rates.