Question

Calculate the value of x for the cash flows below such that the equivalent total value in year 8 is $20,000, using an interest rate of 15% per year.

Year 1: $5,000
Year 2: $4,000
Year 3: $3,000
Year 4: $2,000
Year 5: $1,000
Year 6: $500
Year 7: $200
Year 8: $x

Answer 1

To find the value of x for the cash flows, discount each cash flow to its present value at year 8 using the formula provided, sum these values, and solve for x such that the total equals $20,000 with a 15% annual interest rate.

Step-by-step explanation:

Calculating the Missing Cash Flow

To calculate the value of x for the cash flows so that the equivalent total value in year 8 is $20,000 using an interest rate of 15% per year, we need to discount each cash flow back to its present value at year 8 and ensure the sum equals $20,000. The formula used for discounting is Future value received years in the future = Payment / (1 + Interest rate)number of years t. We apply this to each year's cash flow.

• The present value in year 8 of the $5,000 received in year 1 is $5,000 / (1+0.15)7.
• Continuing for each cash flow, we discount $4,000 from year 2, $3,000 from year 3, and so on.
• We then solve for x, knowing that when all these present values are summed, they must equal $20,000 in year 8.

To find the total value of the cash flows except for x, we discount each amount using the given formula, sum these present values, and subtract from the future value of $20,000 in year 8. The difference will give us the present value of x that when added to the other present values gives the total of $20,000 in year 8.