suppose the interest rate is 8% APR with monthly compounding. what is the present value of the annuity that pays $100 for every 6 years?
To calculate the present value of an annuity, we can use the formula:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value
PMT = Payment per period
r = Interest rate per period
n = Number of periods
In this case, the payment per period is $100, the interest rate per period is 8% / 12 (since it is compounded monthly), and the number of periods is 6 years * 12 months = 72 months.
Substituting the values into the formula:
PV = $100 * (1 - (1 + 0.08/12)^(-72)) / (0.08/12)
Now, let's calculate the present value.
To calculate the present value of an annuity, we need to use the formula:
PV = PMT * (1 - (1 + r)^(-n)) / r
In this formula:
PV = Present Value
PMT = Payment per period (in this case $100)
r = Interest rate per period (in this case 8% APR, which needs to be adjusted for monthly compounding)
n = Number of periods (in this case 6 years)
First, let's adjust the interest rate for monthly compounding. Since the interest rate is given as an annual percentage rate (APR), we need to divide it by 12 to get the monthly interest rate. Therefore, the monthly interest rate will be 8% / 12 = 0.08 / 12 = 0.0067.
Now, we can plug in the values into the formula:
PV = $100 * (1 - (1 + 0.0067)^(-6*12)) / 0.0067
To solve this, we need to calculate the value of (1 + 0.0067)^(-6*12), which represents the compound interest factor for 6 years with monthly compounding.
Using a calculator, we find that (1 + 0.0067)^(-6*12) is approximately 0.56128.
Now we can substitute this value back into the formula:
PV = $100 * (1 - 0.56128) / 0.0067
Simplifying this further:
PV = $100 * (0.43872) / 0.0067
Finally, calculating the present value:
PV ≈ $6607.46
Therefore, the present value of the annuity that pays $100 every 6 years with an 8% APR and monthly compounding is approximately $6607.46.