To clear a debt, a person agrees to pay thousand now, another 1000 a year from now and another thousand in 2 years. If the future payments are discount at 8% compounded quarterly, what is the present value of these payments?
To find the present value of the future payments, we need to discount each payment back to its present value using the given interest rate.
Let's calculate the present value for each payment separately.
First payment of $1000:
Since this payment is made now, its present value is equal to its future value. So, the present value of the first payment is $1000.
Second payment of $1000 [after 1 year]:
To find the present value of this payment, we need to discount it back 1 year at an interest rate of 8% compounded quarterly. Quarterly compounding means that the interest rate per quarter is 8%/4 = 2%.
Using the compound interest formula, the present value (PV) of the second payment can be calculated as:
PV = FV / (1+r)^n
Where FV is the future value, r is the interest rate per period, and n is the number of periods.
In this case, FV = $1000, r = 2% or 0.02, and n = 4 (since there are 4 quarters in a year).
PV = $1000 / (1+0.02)^4
PV = $1000 / (1.02)^4
PV ≈ $924.16
Third payment of $1000 [after 2 years]:
Using the same formula as above, the present value (PV) of the third payment can be calculated as:
PV = FV / (1+r)^n
In this case, FV = $1000, r = 2% or 0.02, and n = 8 (since there are 8 quarters in 2 years).
PV = $1000 / (1+0.02)^8
PV = $1000 / (1.02)^8
PV ≈ $852.57
Lastly, to find the total present value of the payments, we need to sum up the present values of each payment.
Total Present Value = Present Value of the first payment + Present Value of the second payment + Present Value of the third payment
Total Present Value = $1000 + $924.16 + $852.57
Total Present Value ≈ $2776.73
Therefore, the present value of the payments is approximately $2776.73.
To calculate the present value of future payments, we need to discount each payment to its present value using the formula for compound interest:
Present Value = Future Value / (1 + r)^n
Where:
- Future Value is the amount to be paid in the future
- r is the interest rate per compounding period
- n is the number of compounding periods
In this case, the future payments are $1000 each, and the interest rate is 8% compounded quarterly. Let's calculate the present value for each payment and sum them up:
Payment 1 (now):
Present Value 1 = $1000 / (1 + 0.08/4)^0 = $1000 / (1 + 0.02)^0 = $1000
Payment 2 (after 1 year):
Present Value 2 = $1000 / (1 + 0.08/4)^4 = $1000 / (1 + 0.02)^4 = $924.20 (rounded to two decimal places)
Payment 3 (after 2 years):
Present Value 3 = $1000 / (1 + 0.08/4)^8 = $1000 / (1 + 0.02)^8 = $853.18 (rounded to two decimal places)
Now, we can sum up the present values of all three payments:
Present Value Total = Present Value 1 + Present Value 2 + Present Value 3
Present Value Total = $1000 + $924.20 + $853.18
Present Value Total = $2777.38 (rounded to two decimal places)
Therefore, the present value of these three payments is $2777.38.
To find the present value of future payments, we need to discount each payment back to the present using the formula for the present value of a lump sum.
The formula is:
PV = FV / (1 + r/n)^(n*t)
Where:
PV = Present Value
FV = Future Value
r = Interest Rate
n = Number of compounding periods per year
t = Number of years
In this case, the future payments are $1000 each, there are 2 years until the final payment, and the interest rate is 8%. The payments are compounded quarterly, so there are 4 compounding periods per year.
Let's calculate the present value of each payment step-by-step and then sum them up:
1. Present Value of $1000 after 1 year:
PV1 = 1000 / (1 + 0.08/4)^(4*1)
PV1 = 1000 / (1 + 0.02)^4
PV1 = 1000 / (1.02)^4
PV1 ≈ $924.32
2. Present Value of $1000 after 2 years:
PV2 = 1000 / (1 + 0.08/4)^(4*2)
PV2 = 1000 / (1 + 0.02)^8
PV2 = 1000 / (1.02)^8
PV2 ≈ $857.34
3. Present Value of $1000 after 3 years:
PV3 = 1000 / (1 + 0.08/4)^(4*3)
PV3 = 1000 / (1 + 0.02)^12
PV3 = 1000 / (1.02)^12
PV3 ≈ $793.83
Now, let's sum up the present values of all three payments to find the total present value of the debt:
Total Present Value = PV1 + PV2 + PV3
Total Present Value ≈ $924.32 + $857.34 + $793.83
Total Present Value ≈ $2,575.49
Therefore, the present value of these payments is approximately $2,575.49.