To find the present value of an annuity, we can use the formula for the present value of an ordinary annuity, which is given by:
\[ PV = P \times \left( \frac{1 - (1 + r)^{-n}}{r} \right) \]
Where:
- \(PV\) = Present Value
- \(P\) = Payment per period
- \(r\) = Interest rate per period
- \(n\) = Total number of payments
In this case:
- The annual interest rate is 6%, so the monthly interest rate \(r\) is \( \frac{6%}{12} = 0.5% = 0.005 \).
- The payment per month \(P\) is $100.
- The total number of payments \(n\) is 12 (1 year of monthly payments).
Now we can substitute these values into the formula:
\[ PV = 100 \times \left( \frac{1 - (1 + 0.005)^{-12}}{0.005} \right) \]
Calculating further:
- Calculate \(1 + r\):
\[ 1 + 0.005 = 1.005 \]
- Calculate \((1 + r)^{-n}\):
\[ (1.005)^{-12} \approx 0.9409 \]
- Now substitute back into the formula:
\[ PV = 100 \times \left( \frac{1 - 0.9409}{0.005} \right) \]
- Calculate \(1 - 0.9409\):
\[ 1 - 0.9409 = 0.0591 \]
- Now compute the fraction:
\[ \frac{0.0591}{0.005} \approx 11.82 \]
- Finally multiply by 100:
\[ PV = 100 \times 11.82 \approx 1182 \]
Thus, the present value of the annuity is approximately $1182.00.