Suppose the interest rate is 7.9% APR with monthly compounding. What is the present value of an annuity that pays $80 every six months for four ​years?

To find the present value of an annuity that pays $ 80 $ dollars every six months over four years with a monthly compounded annual interest rate of $ 7.9\% $, we can use the following steps:

### Step 1: Determine the number of payment periods.
The annuity pays every six months for four years, so there are:

\[
\text{Number of periods} = 4 \, \text{years} \times 2 \, \text{periods/year} = 8 \, \text{periods}.
\]

### Step 2: Convert the APR to the effective interest rate per period.
Since the APR is $ 7.9\% $ with monthly compounding, we need to find the semi-annual interest rate. First, we convert the monthly rate to a semi-annual rate.

The monthly interest rate $ i_{\text{monthly}} $ is:

\[
i_{\text{monthly}} = \frac{7.9\%}{12} = \frac{0.079}{12} \approx 0.0065833 \, \text{per month}.
\]

Now, to find the semi-annual interest rate, we compound this over 6 months:

\[
i_{\text{semi-annual}} = (1 + i_{\text{monthly}})^{6} - 1 = (1 + 0.0065833)^6 - 1 \approx 0.040572.
\]

Thus, the effective semi-annual interest rate is $ 4.0572\% $ or $ 0.040572 $.

### Step 3: Use the present value of an annuity formula.
The formula for the present value $ PV $ of an ordinary annuity is:

\[
PV = P \times \frac{1 - (1 + r)^{-n}}{r},
\]

where:
- $ P = 80 $ is the payment per period,
- $ r = 0.040572 $ is the semi-annual interest rate,
- $ n = 8 $ is the total number of periods.

Now, substitute the values:

\[
PV = 80 \times \frac{1 - (1 + 0.040572)^{-8}}{0.040572}.
\]

### Step 4: Calculate the present value.
First, calculate $ (1 + 0.040572)^{-8} $:

\[
(1 + 0.040572)^{-8} \approx 0.70707.
\]

Now, substitute this back into the equation:

\[
PV = 80 \times \frac{1 - 0.70707}{0.040572} = 80 \times \frac{0.29293}{0.040572} \approx 80 \times 7.220 = 577.6.
\]

Thus, the present value of the annuity is approximately **$577.60**.

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Do you need more details or have any questions about the solution?

Here are 5 related questions to expand on the topic:
1. What happens to the present value if the interest rate increases to 10% APR?
2. How would the present value change if the annuity paid monthly instead of every six months?
3. What is the future value of the annuity after four years with the same interest rate?
4. How does the present value formula differ for continuous compounding of interest?
5. Can you calculate the total interest earned if the annuity pays out the same amount for ten years?

**Tip:** When dealing with APRs, always adjust the interest rate to match the payment frequency for accurate calculations.

Learn how to calculate the present value of an annuity that pays $80 every six months for four years, with an interest rate of 7.9% APR and monthly compounding. This step-by-step solution includes converting the interest rate to semi-annual terms and applying the present value formula. Suitable for high school students in Grades 11-12.

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