Find the present value of the ordinary annuity. (Round your answer to the nearest cent.)
$5000/semiannual period for 8 years at 3%/year compounded semiannually
$ 


To find the present value of an ordinary annuity, we use the formula:

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

where:
- $ PV $ is the present value of the annuity.
- $ P $ is the payment amount per period.
- $ r $ is the interest rate per period.
- $ n $ is the total number of payments.

### Given:
- $ P = 5000 $ (payment per semiannual period)
- The annual interest rate is 3%, so the semiannual interest rate $ r $ is $ \frac{3\%}{2} = 1.5\% = 0.015 $.
- The total number of years is 8, and since payments are made semiannually, the total number of periods $ n $ is $ 8 \times 2 = 16 $.

### Plugging in the values:
\[
PV = 5000 \times \left( \frac{1 - (1 + 0.015)^{-16}}{0.015} \right)
\]

Calculating step by step:
1. $ 1 + 0.015 = 1.015 $
2. $ 1.015^{-16} \approx 0.7835 $
3. $ 1 - 0.7835 = 0.2165 $
4. $ \frac{0.2165}{0.015} \approx 14.4333 $

Finally, multiplying by $ P $:
\[
PV = 5000 \times 14.4333 \approx 72166.67
\]

Thus, the present value of the annuity is approximately **$72,166.67**.

Would you like to go through any particular step in more detail? 

Here are five related questions:
1. What is the present value if the interest rate was 4% annually, compounded semiannually?
2. How does compounding frequency affect the present value of an annuity?
3. What would be the future value of this annuity after 8 years?
4. How would the present value change if the payments were made monthly instead of semiannually?
5. What is the difference in present value between an ordinary annuity and an annuity due?

**Tip:** Increasing the compounding frequency (e.g., semiannual to quarterly) generally results in a higher present value for a given annuity.

Find the present value of the ordinary annuity. (Round your answer to the nearest cent.) $5000/semiannual period for 8 years at 3%/year compounded semiannually.

Learn how to calculate the present value of an ordinary annuity with $5000 semiannual payments for 8 years at an annual interest rate of 3%, compounded semiannually. This detailed step-by-step guide uses the annuity present value formula, perfect for advanced high school or college students.

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