To calculate the present value of a series of future cash flows (an annuity), we can use the present value formula:
\[ PV = P \times \left(1 - (1 + r)^{-n}\right) / r \]
Where:
- \( PV \) is the present value of the annuity.
- \( P \) is the annual payment ($4,855.00).
- \( r \) is the interest rate (6.00% or 0.06).
- \( n \) is the number of payments (16.00 years).
Since the first payment starts in 2 years, we need to calculate the present value at the time of the first payment, and then discount that value back to today.
- Calculate the present value of the annuity at the time of the first payment (in 2 years):
Using the formula for present value of an annuity:
\[ PV_{2} = 4855 \times \left(1 - (1 + 0.06)^{-16}\right) / 0.06 \]
Calculating the values:
\[ 1 + 0.06 = 1.06 \] \[ (1.06)^{-16} = \frac{1}{(1.06)^{16}} \approx 0.3956 \quad \text{(using a calculator)} \] \[ 1 - 0.3956 = 0.6044 \] Now we can plug values into the annuity formula:
\[ PV_{2} = 4855 \times \frac{0.6044}{0.06} \] \[ PV_{2} = 4855 \times 10.0733 \approx 48,908.81 \]
- Now, we need to discount this amount back to today (from year 2 to year 0).
The formula for present value is:
\[ PV_{0} = PV_{2} \times (1 + r)^{-t} \]
Where \( t \) is the number of years until the first payment (which is 2 years):
\[ PV_{0} = 48,908.81 \times (1.06)^{-2} \]
Calculating \( (1.06)^{-2} \):
\[ (1.06)^{-2} = \frac{1}{1.1236} \approx 0.8890 \quad \text{(using a calculator)} \] Now calculate:
\[ PV_{0} = 48,908.81 \times 0.8890 \approx 43,426.55 \]
Thus, the present value today of the annuity that pays $4,855.00 per year for 16 years, with the first payment starting in 2 years, at an interest rate of 6% is approximately $43,426.55.
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