9. Compute the present value of $4,000 paid in five years using the following discount rates: 10 percent in year 1, 2 percent in year 2, 12 percent in year 3, and 9 percent in years 4 and 5.
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To compute the present value of $4,000 paid in five years using the given discount rates, we will use the formula for the present value of a future payment:
PV = FV / (1+r)^n
Where PV is the present value, FV is the future value, r is the discount rate, and n is the number of years.
Let's calculate the present value step by step for each discount rate:
1. Year 1 discount rate is 10%:
PV1 = $4,000 / (1 + 0.10)^1
= $4,000 / 1.10
= $3,636.36
2. Year 2 discount rate is 2%:
PV2 = $4,000 / (1 + 0.02)^2
= $4,000 / 1.0404
= $3,843.25
3. Year 3 discount rate is 12%:
PV3 = $4,000 / (1 + 0.12)^3
= $4,000 / 1.404928
= $2,846.05
4. Year 4 discount rate is 9%:
PV4 = $4,000 / (1 + 0.09)^4
= $4,000 / 1.411581
= $2,834.08
5. Year 5 discount rate is 9%:
PV5 = $4,000 / (1 + 0.09)^5
= $4,000 / 1.538624
= $2,603.84
Therefore, the present value of $4,000 paid in five years using the given discount rates is the sum of the present values calculated above:
PV = PV1 + PV2 + PV3 + PV4 + PV5
= $3,636.36 + $3,843.25 + $2,846.05 + $2,834.08 + $2,603.84
= $15,763.58
To compute the present value of $4,000 paid in five years using the given discount rates, we need to find the present value of each cash flow and then sum them up.
1. Start with the cash flow in year 1, which is $4,000. Since the discount rate is 10% in year 1, we need to discount this cash flow back to present value. The formula to calculate the present value is:
PV = CF / (1 + r)^n
where PV is the present value, CF is the cash flow, r is the discount rate, and n is the number of years.
Using this formula, we can calculate the present value of the cash flow in year 1:
PV1 = $4,000 / (1 + 0.10)^1
2. For the cash flow in year 2, the discount rate is 2%. We can use the formula again to find the present value:
PV2 = $4,000 / (1 + 0.02)^2
3. For the cash flow in year 3, the discount rate is 12%. Find the present value using the formula:
PV3 = $4,000 / (1 + 0.12)^3
4. For the cash flows in years 4 and 5, the discount rate is 9%. Calculate the present value for both years separately using the formula:
PV4 = $4,000 / (1 + 0.09)^4
PV5 = $4,000 / (1 + 0.09)^5
5. Finally, we can find the total present value by summing up all the individual present values:
Total PV = PV1 + PV2 + PV3 + PV4 + PV5
Using the given discount rates, substitute them into the formulas for each year and calculate the present values. Finally, add up all the present values to get the total present value of $4,000 paid in five years.