To clear a net a person agrees to pay 1000 now another 1000 a Year from now and another 1000 in 2 years. if the future payments are discounted at 8% compounded quarterly, what is the present value of these payment?
To find the present value of future payments, we need to discount each payment back to the present value using the given discount rate.
The payment of 1000 now has already been discounted at 0%, so its present value remains 1000.
The payment of 1000 in 1 year needs to be discounted at a rate of 8% compounded quarterly. The general formula to calculate the present value of a future payment using compound interest is:
PV = FV / (1 + r/n)^(n*t)
Where PV is the present value, FV is the future value, r is the interest rate, n is the number of compounding periods per year, and t is the number of years.
Plugging in the given values:
PV = 1000 / (1 + 0.08/4)^(4*1)
PV = 1000 / (1 + 0.02)^4
PV = 1000 / (1.02)^4
PV = 1000 / 1.082432
PV ≈ 923.54
Therefore, the present value of the payment of 1000 in 1 year is approximately 923.54.
The payment of 1000 in 2 years needs to be discounted at the same rate of 8% compounded quarterly. Using the same formula:
PV = 1000 / (1 + 0.08/4)^(4*2)
PV = 1000 / (1 + 0.02)^8
PV = 1000 / (1.02)^8
PV = 1000 / 1.171665
PV ≈ 852.13
Therefore, the present value of the payment of 1000 in 2 years is approximately 852.13.
Finally, to find the present value of all the payments, we sum up the present values:
Present Value = 1000 + 923.54 + 852.13
Present Value ≈ 2775.67
Therefore, the present value of these payments is approximately 2775.67.
To find the present value of future payments, we need to discount each payment back to its present value. Let's calculate the present value of each payment step-by-step.
Step 1: Calculate the present value of the first payment of 1000 now.
Since this payment is at the present time, there is no need to discount it.
Present Value (PV1) = 1000
Step 2: Calculate the present value of the second payment of 1000 in one year.
The formula to calculate the present value (PV) using quarterly compounding is:
PV = Payment / (1 + r/n)^(n*t)
Where:
PV = Present value
Payment = Future payment
r = Interest rate
n = Number of compounding periods per year
t = Number of years
Using the given values:
Payment = 1000
r = 8% = 0.08
n = 4 (quarterly compounding)
t = 1 (1 year)
PV2 = 1000 / (1 + 0.08/4)^(4*1)
PV2 = 1000 / (1.02)^(4)
PV2 ≈ 941.65
Step 3: Calculate the present value of the third payment of 1000 in two years.
Using the same formula as in Step 2, but with t = 2:
PV3 = 1000 / (1 + 0.08/4)^(4*2)
PV3 = 1000 / (1.02)^(8)
PV3 ≈ 885.55
Step 4: Calculate the total present value by summing up the present values of all three payments.
Total present value = PV1 + PV2 + PV3
Total present value = 1000 + 941.65 + 885.55
Total present value ≈ 2827.20
Therefore, the present value of the three payments is approximately 2827.20.