A 5-year annuity of ten $4500 semiannual payments will begin 9 years from now, with the first payment coming 9.5 years from now. If the discount rate is 12% compounded monthly, what is the value of this annuity five years from now? What is the value three years from now? What is the current value of the annuity?
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To calculate the value of the annuity five years from now, three years from now, and the current value, we will use the present value of an ordinary annuity formula.
The present value of an ordinary annuity formula is given by:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value
PMT = Payment per period
r = Interest rate per period
n = Total number of periods
Given:
Payment per period (PMT) = $4500
Interest rate per period (r) = 12%/12 = 1% (since the discount rate is compounded monthly)
Total number of periods (n) = 10
1. Calculation for the value of the annuity five years from now:
The annuity starts 9.5 years from now, so the total number of periods for this calculation is 10 - (9.5/0.5) = 10 - 19 = -9. (Negative because we are working backward in time)
PV = $4500 * (1 - (1 + 0.01)^(-9)) / 0.01
PV = $4500 * (1 - (1.01)^(-9)) / 0.01
PV ≈ $28,205.69
The value of the annuity five years from now is approximately $28,205.69.
2. Calculation for the value of the annuity three years from now:
The annuity starts 9.5 years from now, so the total number of periods for this calculation is 10 - (9.5/0.5) + 2 = 10 - 19 + 2 = -7.
PV = $4500 * (1 - (1 + 0.01)^(-7)) / 0.01
PV = $4500 * (1 - (1.01)^(-7)) / 0.01
PV ≈ $16,384.27
The value of the annuity three years from now is approximately $16,384.27.
3. Calculation for the current value of the annuity:
The annuity starts 9.5 years from now, so the total number of periods for this calculation is 10 - (9.5/0.5) + 5 = 10 - 19 + 5 = -4.
PV = $4500 * (1 - (1 + 0.01)^(-4)) / 0.01
PV = $4500 * (1 - (1.01)^(-4)) / 0.01
PV ≈ $11,112.29
The current value of the annuity is approximately $11,112.29.
To find the value of the annuity, we can use the formula for the present value of an annuity:
PV = PMT * [(1 - (1 + r)^(-n)) / r]
Where:
PV = Present value of the annuity
PMT = Payment amount ($4500)
r = Discount rate per period (12% per year compounded monthly, so 12%/12 = 1% per month)
n = Total number of payment periods
First, let's calculate the total number of payment periods.
Since there are 10 semiannual payments (twice a year), and the annuity begins 9 years from now, the total number of payment periods would be (10 * 2) = 20 payments.
Now let's calculate the present value of the annuity five years from now.
Step 1: Determine the total number of payment periods from the point in time where we want to calculate the present value.
Since the annuity starts 9 years from now and we want to calculate the value five years from now, the total number of payment periods would be (9 - 5) * 2 = 8.
Step 2: Calculate the present value.
PV = PMT * [(1 - (1 + r)^(-n)) / r]
PV = $4500 * [(1 - (1 + (1%/100))^(-8)) / (1%/100)]
PV = $4500 * [(1 - (1.01)^(-8)) / 0.01]
Now, we can calculate the value three years from now using the same method as above:
Step 1: Determine the total number of payment periods from the point in time where we want to calculate the present value.
Since the annuity starts 9 years from now and we want to calculate the value three years from now, the total number of payment periods would be (9 - 3) * 2 = 12.
Step 2: Calculate the present value.
PV = PMT * [(1 - (1 + r)^(-n)) / r]
PV = $4500 * [(1 - (1 + (1%/100))^(-12)) / (1%/100)]
PV = $4500 * [(1 - (1.01)^(-12)) / 0.01]
Finally, to calculate the current value of the annuity, we need to calculate the present value from the point when the annuity starts.
Step 1: Determine the total number of payment periods.
Since the annuity starts 9 years from now, the total number of payment periods would be (9 * 2) = 18.
Step 2: Calculate the present value.
PV = PMT * [(1 - (1 + r)^(-n)) / r]
PV = $4500 * [(1 - (1 + (1%/100))^(-18)) / (1%/100)]
PV = $4500 * [(1 - (1.01)^(-18)) / 0.01]
By plugging in the values and performing the calculations, you can find the specific present values.