A 15-year annuity pays $1,750 per month, and payments are made at the end of each month. If the interest rate is 10 percent compounded monthly for the first seven years, and 6 percent compounded monthly thereafter, what is the present value of the annuity?
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To calculate the present value of the annuity, we need to break it down into two parts: the first seven years with a 10 percent interest rate, and the remaining years with a 6 percent interest rate.
Step 1: Calculate the present value of the annuity for the first seven years:
Using the formula for the present value of an ordinary annuity:
PV = PMT * [(1 - (1 + r)^(-n)) / r]
Where:
PMT = monthly payment = $1,750
r = interest rate per period = 10% / 12 = 0.8333%
n = number of periods = 7 * 12 = 84 (as there are 12 months in a year)
By plugging in the values into the formula:
PV_1 = $1,750 * [(1 - (1 + 0.008333)^(-84)) / 0.008333]
Step 2: Calculate the present value of the annuity for the remaining years:
To calculate the present value of an annuity with a different interest rate, we need to use a more general formula:
PV_2 = PMT * [(1 - (1 + r)^(-n)) / r]
Where:
PMT = monthly payment = $1,750
r = interest rate per period = 6% / 12 = 0.5%
n = number of periods = (15 - 7) * 12 = 96 (as there are 12 months in a year, and we subtract 7 years from the total)
By plugging in the values into the formula:
PV_2 = $1,750 * [(1 - (1 + 0.005)^(-96)) / 0.005]
Step 3: Calculate the total present value of the annuity:
PV_total = PV_1 + PV_2
By substituting the calculated values:
PV_total = PV_1 + PV_2
Note: Without the actual values for PMT and interest rates, it's not possible to calculate the exact present value.
To find the present value of the annuity, we need to determine the value of all the future cash flows and discount them back to the present value using the appropriate interest rates for each period.
Step 1: Calculate the present value of the first seven years.
Since the interest rate for this period is 10%, compounded monthly, we can use the formula for the present value of a future value (PV) of a monthly payment (PMT) over a certain number of periods (n) at a given interest rate (r).
PV = PMT * (1 - (1 + r)^(-n)) / r
Calculating PV for the first seven years:
PMT = $1,750
r = 10% / 12 (monthly interest rate)
n = 7 * 12 (number of months)
PV1 = $1,750 * (1 - (1 + (10% / 12))^(-7 * 12)) / (10% / 12)
Step 2: Calculate the present value of the remaining eight years.
Since the interest rate for this period is 6%, compounded monthly, we can use the same formula as above, replacing the interest rate and the number of months.
Calculating PV for the remaining eight years:
PMT = $1,750
r = 6% / 12 (monthly interest rate)
n = 8 * 12 (number of months)
PV2 = $1,750 * (1 - (1 + (6% / 12))^(-8 * 12)) / (6% / 12)
Step 3: Calculate the total present value by summing the present value of the first seven years and the present value of the remaining eight years.
Total Present Value = PV1 + PV2
Once you have calculated the values for PV1 and PV2, simply add them together to find the total present value of the annuity.