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To calculate the **present value** and **compound interest**, we will use the compound interest formula:

\[
A = P \left( 1 + \frac{r}{n} \right)^{nt}
\]

Where:
- $ A $ is the **future value** or **compound amount** ($500,000 in this case),
- $ P $ is the **present value** (what we are solving for),
- $ r $ is the **annual nominal interest rate** (4% or 0.04 in this case),
- $ n $ is the number of times the interest is compounded per year (annually, so $ n = 1 $),
- $ t $ is the number of years (10 years in this case).

Rearranging the formula to solve for $ P $:

\[
P = \frac{A}{\left( 1 + \frac{r}{n} \right)^{nt}}
\]

### Substituting the values:
- $ A = 500,000 $,
- $ r = 0.04 $,
- $ n = 1 $,
- $ t = 10 $,

We get:

\[
P = \frac{500,000}{\left( 1 + \frac{0.04}{1} \right)^{1 \times 10}} = \frac{500,000}{\left( 1.04 \right)^{10}}
\]

Let's calculate:

\[
P = \frac{500,000}{1.48024} \approx 337,843.59
\]

So, the **Present Value** is approximately **$337,843.59**.

### Compound Interest:
Now, we can calculate the compound interest:

\[
\text{Compound Interest} = A - P = 500,000 - 337,843.59 = 162,156.41
\]

Thus, the **Compound Interest** is approximately **$162,156.41**.

---

Do you want further details or clarifications? Here are some follow-up questions:

1. How does changing the compounding frequency affect the present value?
2. What would the present value be if the interest rate increased to 5%?
3. How can you calculate compound interest for monthly compounding instead of annually?
4. What would happen to the compound interest if the term was 15 years instead of 10?
5. How does inflation impact the interpretation of compound interest over time?

**Tip:** Always check if interest is compounded more frequently than annually, as this can significantly affect the final amount!

Calculate the present value and compound interest for a compound amount of $500,000 over 10 years at an annual nominal rate of 4%, compounded annually.

Learn how to calculate the present value and compound interest for a compound amount of $500,000 invested for 10 years at a nominal interest rate of 4%, compounded annually. Includes step-by-step breakdowns of the compound interest and present value formulas.

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