Question

Find the present value of an annuity of $100 payable at the end of each month for 1 year at 6% p.a interest

Answer 1

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:

1. Calculate \(1 + r\):

\[
1 + 0.005 = 1.005
\]

2. Calculate \((1 + r)^{-n}\):

\[
(1.005)^{-12} \approx 0.9409
\]

3. Now substitute back into the formula:

\[
PV = 100 \times \left( \frac{1 - 0.9409}{0.005} \right)
\]

4. Calculate \(1 - 0.9409\):

\[
1 - 0.9409 = 0.0591
\]

5. Now compute the fraction:

\[
\frac{0.0591}{0.005} \approx 11.82
\]

6. Finally multiply by 100:

\[
PV = 100 \times 11.82 \approx 1182
\]

Thus, the present value of the annuity is approximately **$1182.00**.